Remark 1 (Different utility functions) 
Our proposed UWMMSE is well suited to solve the optimal power allocation problem for objective functions that go beyond the sum-rate utility.
Adopting the notation of a generic utility \(\beta _i\)  in (REF ), UWMMSE can be adapted to this more encompassing setting by incorporating two changes.
First, the loss function in (REF ) must be updated to \(\ell (\Theta ) = - \mathbb {E}_{{\mathbf {H}}\sim {\mathcal {H}}} \left[ \sum _{i=1}^M \beta _i(c_i(\Phi ({\mathbf {H}}; \Theta ), {\mathbf {H}})) \right]\) .
Second, and more noteworthy, we must adapt the unfolded architecture to correspond to the WMMSE solution of the modified utility problem.
This is unlike state-of-the-art methods [1]}, [2]}, [3]} where the architecture is unaffected by the utility at hand, thus indicating a limited incorporation of domain knowledge.
More precisely, defining the scalar functions \(\gamma _i(z) = - \beta _i (-\log (z))\) , the update in () must be changed to
\(w_i^{(k)} = \gamma _i^{\prime }(1 - u^{(k)}_i h_{ii} v^{(k-1)}_i) \, a_i^{(k)} + b_i^{(k)},\) 
